In the presence of KC, rapid cooling mechanisms through \(\nu \) , \(\bar{\nu }\)  emissions may be kinematically possible, which plays an important role on thermal evolution of neutron stars: One is the kaon-induced Urca (KU) process, \(N + \langle K^-\rangle \rightarrow N + e^- +\bar{\nu }_e\) , \(N + e^- \rightarrow N + \langle K^- \rangle + \nu _e\)  (\(N=p,n\) ), where \(\langle K^- \rangle \)  stands for the classical \(K^-\)  field which supplies the system with energy \(\mu _K\)  to make the reaction kinematically possible [1]}, [2]}. The other is the direct Urca (DU) process in KC, \(n\rightarrow p+e^- +\bar{\nu }_e\) , \(p+e^-\rightarrow n + \nu _e\) , as long as the kinematical condition for the reaction is met depending upon the density-dependence of the symmetry energy [3]}.
According to the results in the present work,
main cooling process is divided by the mass \(M \sim 1.4 M_\odot \) . For \(M\lesssim 1.4 M_\odot \) , it is given by the modified Urca process since the proton-mixing ratio is under threshold for the DU process, \(\rho _p/\rho _{\rm B}\lesssim 1/9\) .
For \(M > 1.4 M_\odot \) , Hyperon (\(\Lambda \) ) Urca process, \(\Lambda \rightarrow p+e^- +\bar{\nu }_e\) , \(p+e^-\rightarrow \Lambda + \nu _e\)  [4]} starts and becomes a dominant cooling process. For the massive neutron stars (\(M \gtrsim 1.7~M_\odot \) ), the KU process becomes a main cooling process.
Several neutron stars have anomalously low temperature that requires extraordinary rapid cooling processes [5]}, [6]}. Unified description of emissivities for these reactions with composition of matter is indispensable.
